Necessary conditions for an extremum in a mathematical programming problem

Avakov, E. R.; Arutyunov, A. V.; Izmailov, A. F.

doi:10.1134/s0081543807010014

Cited by 19 publications

(14 citation statements)

References 7 publications

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“…The obtained results are based on the general theory developed in [1] and they generalize our recent results obtained in [4] to the case of discrete optimal control problems with equality and inequality type of constraints on control and on endpoints.…”

Section: Introductionmentioning

confidence: 67%

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Optimality conditions for discrete optimal control problems with equality and inequality type of constraints

Marinković¹

2008

Positivity

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Section: Introductionmentioning

confidence: 67%

“…In theorem 3 from [1] was proved that there exists Lagrange multiplierλ such that for every (h, v) ∈C 1 2 (ξ,ŵ) the following conditions hold:…”

Section: B Marinković Positivitymentioning

confidence: 99%

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Optimality conditions for discrete optimal control problems with equality and inequality type of constraints

Marinković¹

2008

Positivity

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“…Substituting y * 1 = x 3 1 into (54), we get y * 2 = 2x 3 1 . From x 2 = x 1 + u 1 = x 1 , y * 2 = 2x 3 1 and (55), we have 4x 2 (1 + x 2 2 ) 2 = 2x 3 2 .…”

Section: Some Examplesmentioning

confidence: 97%

Second-order necessary optimality conditions for a discrete optimal control problem with mixed constraints

Toan

Thuy

2015

J Glob Optim

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“…There also exists some literature on numerical methods to solve this type of problems [2,5,15]. The two last papers cover mixed (equality and inequality) constraints.…”

Section: Introductionmentioning

confidence: 99%

Generalized convexity in non-regular programming problems with inequality-type constraints

Hernández‐Jiménez

Rojas-Medar

Osuna-Gómez

et al. 2009

Journal of Mathematical Analysis and Applications

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Convexity plays a very important role in optimization for establishing optimality conditions. Different works have shown that the convexity property can be replaced by a weaker notion, the invexity. In particular, for problems with inequality-type constraints, Martin defined a weaker notion of invexity, the Karush-Kuhn-Tucker-invexity (hereafter KKTinvexity), that is both necessary and sufficient to obtain Karush-Kuhn-Tucker-type optimality conditions. It is well known that for this result to hold the problem has to verify a constraint qualification, i.e., it must be regular or non-degenerate. In non-regular problems, the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with inequality-type constraints by Izmailov. They are based on the 2-regularity condition of the constraints at a feasible point. In this work, we generalize Martin's result to non-regular problems by defining an analogous concept, the 2-KKTinvexity, and using the characterization of the tangent cone in the 2-regular case and the necessary optimality condition given by Izmailov.

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Necessary conditions for an extremum in a mathematical programming problem

Cited by 19 publications

References 7 publications

Optimality conditions for discrete optimal control problems with equality and inequality type of constraints

Optimality conditions for discrete optimal control problems with equality and inequality type of constraints

Second-order necessary optimality conditions for a discrete optimal control problem with mixed constraints

Generalized convexity in non-regular programming problems with inequality-type constraints

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